Editing False (logic) (section)

== In classical logic and Boolean logic ==<!-- linked from #Consistency -->
In [[Boolean logic]], each variable denotes a [[truth value]] which can be either true (1), or false (0).

In a [[classical logic|classical]] [[propositional calculus]], each [[proposition]] will be assigned a truth value of either true or false.<ref>Aristotle (Organon)</ref><ref>Barnes’ Complete Works of Aristotle (Princeton, 1984, Vol. 1):

De Int.: pp. 25–38; Metaphysics IV: pp. 1588–1595.</ref> Some systems of classical logic include dedicated symbols for false (0 or <math>\bot</math>), while others instead rely upon formulas such as {{math|{{mvar|p}} ∧ ¬{{mvar|p}}}} and {{math|¬({{mvar|p}} → {{mvar|p}})}}.

In both Boolean logic and Classical logic systems, true and false are opposite with respect to [[negation]]; the negation of false gives true, and the negation of true gives false.

=== [[Truth table|Truth Tables]] ===
Sources:<ref>Gottlob Frege (1879, Begriffsschrift)</ref><ref>Alfred Tarski (1930s, Introduction to Logic, Chapter II (Symbolic Logic))</ref>

==== Negation (¬) ====

{| class="wikitable" style="text-align: center"
|-
!<math>x</math>
!<math>\neg x</math>
|-
!true
|false
|-
!false
|true
|}

The negation of false is equivalent to the truth not only in classical logic and Boolean logic, but also in most other logical systems, as explained below.

==== Conjunction (AND ∧) ====

False ∧ True = False (False AND anything is False).

==== Disjunction (OR ∨) ====

False ∨ True = True (OR is True if at least one operand is True).

==== Implication (→) ====

False → True = True (A false premise makes the implication vacuously true).